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Some Common Fixed Point Results for Generalized Weak C-contractions in Ordered Metric Spaces



V. Parvaneh and H. Hosseinzadeh
 
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ABSTRACT

This study has presented some common fixed point results for classes of contractions in partially ordered metric spaces. The results has extended and improved the results of several other well-known studies. It also provide the examples to illustrate the results.

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  How to cite this article:

V. Parvaneh and H. Hosseinzadeh, 2012. Some Common Fixed Point Results for Generalized Weak C-contractions in Ordered Metric Spaces. Journal of Applied Sciences, 12: 848-855.

DOI: 10.3923/jas.2012.848.855

URL: https://scialert.net/abstract/?doi=jas.2012.848.855
 
Received: February 04, 2012; Accepted: April 24, 2012; Published: June 27, 2012



INTRODUCTION

Fixed point theory is an interesting field of mathematics. One of its fundamental theorems is Banach's contraction principle (Banach, 1922). This famous result is concerning with the existence and uniqueness of fixed point for contraction mappings, defined on a complete metric space. Alber and Guerre-Delabriere (1997) introduced the concept of weak contraction and after this more attention was devoted to this branch of mathematics. In this direction, development of fixed point theory in partially ordered metric spaces is considerable.

For a survey of fixed point theory, its applications and related results in partially ordered metric spaces we refer to Ran and Reurings (2004), Radenovic and Kadelburg (2010), Nieto and Lopez (2005), Nashine and Samet (2011), Harjani et al. (2011), Abbas et al. (2011), Zhang and Song (2009), Moradi et al. (2011), Doric (2009) and Mujahid and Dragan (2010).

PRELIMINARIES

The concept of C-contraction was introduced by Chatterjea (1972) as follows.

Definition 1: Let (X, d) be a metric space. A mapping T: X→X is said to be a C-contraction if there exists α ε(0, 1/2) such that for all x, yεX the following inequality holds:

Chatterjea (1972) proved that if X is complete, then every C-contraction on X has a unique fixed point.

Choudhury (2009) generalized the concept of C-contraction to weak C-contraction as follows.

Definition 2 : Let (X, d) be a metric space. A mapping T: X→X is said to be weakly C-contractive (or a weak C-contraction) if for all x, yεX:

where, φ: (0, ∞)2→(0, ∞) is a continuous function such that φ (x, y) = 0 if and only if x = y = 0.

Harjani et al. (2011) have presented some fixed point results for weakly C-contractive mappings in a complete metric space endowed with a partially order. One of this results is the following.

Theorem 1: Let (X, ≤) be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let T: X→X be a continuous and nondecreasing mapping such that:

for x≥y, where, φ: (0, ∞)2→(0, ∞) is a continuous function such that φ (x, y) = 0 if and only if x = y = 0. If there exists x0εX with x0≤Tx0, then T has a fixed point.

Moreover, they have proved that the above theorem is still valid for T not necessarily continuous, assuming the following hypothesis:

If (xn) is a nondecreasing sequence in X such that:

(1)

The partially ordered metric spaces with the above property was called regular (Nashine and Samet, 2011).

The notion of an altering distance function was introduced by Khan et al. (1984) as follows.

Definition 3: The function : [0, ∞)→[0, ∞) is called an altering distance function, if the following properties are satisfied.

is continuous and nondecreasing.
(t) = 0 if and only if t = 0.

Let T be a self-map on a metric space X. Beiranvand et al. (2009) introduced the concept of T-contraction mapping as a generalization of the concept of Banach contraction mapping.

A mapping f: X→X is said to be a T-contraction, if there exists a number k in [0,1) such that:

for all x, y in X.

If T = 1 (the identity mapping on X), then the above notion reduces to the Banach contraction mapping.

Definition 4: Let (X, d) be a metric space. A mapping f: X→X is said to be sequentially convergent (subsequentially convergent) if for a sequence {xn} in X for which {fxn} is convergent, {xn} also is convergent ({xn} has a convergent subsequence).

Definition 5: Choudhury and Kundu (2012): Suppose (X, ≤) is a partially ordered set and T, g: X→X are two mappings of X to itself. T is said to be g-non-decreasing if for all x, yεX:

Let denote the class of all altering distance functions : [0, ∞)→[0, ∞) and Φ be the collection of all continuous functions φ: [0, ∞)2→[0, ∞) such that φ (x, y) = 0 if and only if x = y = 0.

Recently, using the concept of an altering distance function, Shatanawi (2011) has presented some fixed point theorems for a nonlinear weakly C-contraction type mapping in metric and ordered metric spaces. His results generalized the results of Harjani et al. (2011).

The following theorems are due to Shatanawi (2011).

Theorem 2: Let (X, ≤, d) be an ordered complete metric space. Let f: X→X be a continuous non-decreasing mapping. Suppose that for comparable x, y, we have:

where, is an altering distance function and φ ε Φ. If there exists x0εX such that x0≤fx0 then f has a fixed point.

Theorem 3: Suppose that X, f, and &phi are as in theorem 1.5 except the continuity of f. Suppose that for a nondecreasing sequence {xn} in X with xn→x ε X, we have xn≤x for all nεN. If there exists x0εX such that x0≤fx0, then f has a fixed point.

Let us note that the beautiful theory of fixed point is used frequently in other branches of mathematics and engineering science (Shakeri, 2009).

The aim of this study is to obtain some common fixed points for weakly C-contractive mappings in a complete and partially ordered complete metric space. Present results extend and generalize the results of Shatanawi (2011), Harjani et al. (2011), Choudhury (2009) and Chatterjea (1972).

MAIN RESULTS

The method of proof has been found by Harjani et al. (2011) and Shatanawi (2011).

Theorem 4: Let (X, ≤, d) be a regular partially ordered complete metric space and T: X→X be an injective, continuous subsequentially convergent mapping. Let f, g: X→X be such that f(X)⊆g(X), f is g-non-decreasing, g(X) is closed and:

(2)

for every pair (x, y)εXxX such that gx≤gy, where, is an altering distance function and φ ε Φ. If there exists x0εX such that gx0≤fx0, then f and g have a coincidence point in X, that is, there exists v ε X such that fv = gv.

Proof: Let x0εX be such that gx0≤fx0. Since f(X)⊆g(X), we can define x1εX such that gx1 = fx0, then gx0≤fx0 = gx1. Since, f is g-non decreasing, we have fx0≤fx1. In this way, we can construct the sequence yn as:

for all n≥0 for which:

Note that, if for all n = 0, 1, ..., we define dn = d (yn, yn+1) and dn = 0 for some n≥0, then yn = yn+1, that is, fxn = gxn+1 = fxn+1 = gxn+2, so g and f have a coincidence point. So, we assume that dn ≠ 0 for each n.

We complete the proof in three steps.

Step 1: We have to prove that:

Using Eq. 2 (which is possible since gxn+1≤gxn ), we obtain that:

(I)

Hence, monotonicity of yields that:

(3)

It follows that the sequence d(Tyn+1, Tyn) is a monotone decreasing sequence of non-negative real numbers and consequently there exists r≥0 such that:

From (I), we have:

If n→∞, we have:

Hence:

We have proved in (I) that:

Now, if n→∞ and since and φ are continuous, we can obtain:

Consequently, φ (0, 2r) = 0. This guarantees that:

Step 2: We show that {Tyn} is a Cauchy sequence in X.

If not, then there exists ε>0 for which we can find subsequences {Tym(k)} and {Tyn(k)} of {Tyn} such that n(k)>m(k)>k and d(Tym(k), Tyn(k))≥ε, where n(k) is the smallest index with this property, i.e.:

(4)

From triangle inequality:

If k→∞, since limn→∞d(Tyn, Tyn+1) = 0, we can conclude that:

(5)

Moreover, we have:

(6)

and

(7)

Since limn→∞ d(Tyn, Tyn+1) = 0 and Eq. 6 and 7 are hold, we get:

(8)

Again, we know that the elements gxm(k) and gxn(k) are comparable (gxn(k)≥gxm(k), as n(k)>m(k)). Putting x = xn(k) and y = xm(k) in Eq. 2, for all k≥0, we have:

If k→∞, from Eq. 4, 8 and the continuity of and φ, we have:

Hence, we have φ (ε, ε) = 0 and therefore, ε = 0 which is a contradiction and it follows that {Tyn} is a Cauchy sequence in X.

Step 3: We show that f and g have a coincidence point.

Since (X, d) is complete and {Tyn} is Cauchy, there exists z ε X such that:

As T is subsequentially convergent, so we have for some u in X where, {fx}ni is a subsequence of {fxn}. Since, T is continuous, which by uniqueness of limit, implies that Tu = z. Since, g(X) is closed and {yni}⊆g(X), we have uεg(X) and hence, there exists vεX such that u = gv.

Now, we prove that v is a coincidence point of f and g.

We know that gxni is a non-decreasing sequence in X such that gxni→u = gv. Thus, from regularity of X, gxni. So, for all I ε , from (2) we have:

If in the above inequality i→∞, we have:

and hence:

and therefore, d(z, Tfv) = 0. So, Tfv = z = Tu. Consequently, fv = u = gv. That is, g and f have a coincidence point.

Theorem 5: Adding the following conditions to the hypotheses of theorem 4, we obtain the existence of the common fixed point of f and g.

(i) gx≤ggx, ∀xεX.
(ii) g and f be weakly compatible.

Moreover, f and g has a unique common fixed point provided that the common fixed points of f and g are comparable.

Proof: We know that gxni = yni-1→u = gv and by our assumptions:

so gxni≤gu and from Eq. 2 we can have:

Since, f and g are weakly compatible and fv = gv, we have fgv = gfv and hence fu = gu.

Now, if i→∞, we obtain:

Hence, φ (d(Tfu, Tu), d(Tu, Tfu)) = 0 and so d(Tfu, Tu) = 0. Therefore, Tfu = Tu. As T is one-to-one, we have fu = u and from fu = gu, we conclude that fu = gu = u.

Let u and v be two common fixed points of f and g, i.e., fu = gu = u and fv = gv = v. Without loss of generality, we assume that u≤v. Then we can apply condition Eq. 2 and obtain:

so, φ (d(Tu, Tv), d(Tv, Tu)) = 0 and hence Tu = Tv. As T is injective, we have u = v.

The following theorem can be proved in a similar way as theorem 4.

Theorem 6: Let (X, ≤, d) be a regular partially ordered complete metric space and T: X→X be an injective, continuous subsequentially convergent mapping. Let f, g: X→X be such that f(X)⊆g(X), f is g-non-decreasing, g(X) is closed and:

(9)

for every pair (x, y) ε XxX such that gx≤gy, where is an altering distance function and φ εΦ.

If there exists x0εX such that gx0≤fx0, then f and g have a coincidence point in X, that is, there exists vεX such that fv = gv.

Moreover, if gx≤ggx, ∀ x ε X and g and f be weakly compatible, then f and g have a common fixed point.

Remark 1: Putting T(x) = g(x) = x (the identity mapping on X) in theorem 4, we obtain the result of Shatanawi (2011) theorem 2 and additionally by taking = I (the identity function on [0, ∞) in theorem 3, we get the result of Harjani et al. (2011), (theorem 1).

Corollary 1: Let (X, ≤, d) be a regular partially ordered complete metric space. Let f, g:X→X be such that f(X)⊆g(X), f is g-non-decreasing, g(X) is closed and:

(10)

for every pair (x, y)εXxX such that gx≥gy, where, is an altering distance function and φ εΦ. If there exists x0 ε X such that gx0≤fx0, then f and g have a coincidence point in X, that is, there exists vεX such that fv = gv.

Moreover, if gx≤ggx, ∀ x ε X and g and f be weakly compatible, then f and g have a common fixed point.

Corollary 1 is a special case of Theorem 3, obtained by setting T = I.

Corollary 2: Let (X, ≤, d) be a regular partially ordered complete metric space and T:X→X be an injective, continuous subsequentially convergent mapping. Let f: X→X is a non-decreasing mapping, and:

(11)

for every pair (x, y) ε XxX such that x≤y, where, is an altering distance function and φ ε Φ. If there exists x0εX such that x0≤fx0, then f has a fixed point in X.

The above Corollary is a special case of Theorem 3, obtained by taking g = I.

The following example support our result.

Example 1: Let X =[0, ∞) be endowed with the usual order and the following metric:

Let T: X→X be defined by Tx = x2, for all xεX. We define functions f: X→X, φ: [0, ∞)2→[0, ∞) and : [0, ∞)→[0, ∞) by:

and (s) = 2s. Then we have:

So, all conditions of theorem 3 are hold. Hence, f and g have a unique common fixed point (x = 0).

Choudhury (2009) proved the following theorem.

Theorem 7: If X is a complete metric space, then every weakly C-contraction T has a unique fixed point (u = Tu for some u ε X).

Now, we go through the four mappings defined on a complete metric space.

Theorem 8: Let (X, d) be a complete metric space and let E be a nonempty closed subset of X. Let T, S: E→E be such that:

(12)

where, ε and φ ε Φ and f, g: E→X are such that:

(A) TE⊆gE and SE⊆fE.
(a) If one of f(E) or g(E) is a closed subspace of X, then g and f and also f and T have a coincidence point.
(b) If S and f as well as T and g are weakly compatible, then f, g, S and T have a unique common fixed point.

Proof: Let x0εE be an arbitrary element. Using (A), there exist two sequences and such that y0 = Tx0 = gx1, y1 = Sx1 = fx2, y2 = Tx2 = gx3, ..., y2n = Tx2n = gx2n+1, y2n+1 = Sx2n+1 = fx2n+2, ....

Note that, if for all n = 0, 1, ..., we define dn = d(yn, yn+1) and d2k = 0 for some n = 2k, then y2k = y2k+1. That is, Tx2k = fx2k+2 = Sx2k+1 = gx2k +1, and so S and g have a coincidence point. Similarly, if d2k+1 = 0, for an n = 2k+1, then f and T have a coincidence point. So, we assume that dn ≠0 for each n. Then, we have the following three steps:

Step I:limn→∞d(yn, yn+1) = 0.

Let n = 2k. Using Eq. 12, we obtain that:

(II)

Hence:

as is nondecreasing.

If n = 2k+1, similarly we can prove that:

Thus, d(yn+1, yn) is a decreasing sequence of nonnegative reals and hence it should be convergent. Let, limn→∞d(yn+1, yn) = r.

From the above argument and in a similar way for n = 2k+1, we have:

and if n→∞, we get:

Therefore:

From (II)

Now, if k→∞ and since and φ are continuous, we can obtain:

and consequently, φ (2r, 0) = 0. This guarantees that:

(13)

from properties of function φ.

Step II:{yn} is Cauchy.

It is enough to show that the subsequence {y2n} is a Cauchy sequence. Suppose that {y2n} is not a Cauchy sequence. Then, there exists ε>0 for which we can find subsequences y2m(k) and y2n(k) of y2n such that n(k)>m(k)>k and:

(14)

and n(k) is the least index with the above property. This means that:

(15)

From Eq. 15 and the triangle inequality:

Letting k→∞ and using Eq. 13 we can conclude that:

(16)

Moreover, we have:

(17)

and:

(18)

Using Eq. 13, 17 and 18, we get:

(19)

Using Eq. 12, we have:

Making k→∞ the above inequality and from Eq. 19 and by the continuity of and φ, we have:

and hence φ (ε, ε) = 0. By our assumption about φ, we have ε = 0 which is a contradiction.

Step III: Existence of coincidence point and common fixed point.

Since (X, d) is complete and {yn} is Cauchy, there exists zεX such that limn→∞y2n = fx2n = z. Since, E is closed and {yn}⊆E, we have z ε E. If we assume that f(E) is closed, then there exists u εE such that z = fu.

Form (12), we see that:

Now, if n→∞.

and hence:

and therefore, d(z, Tu) = 0. So, Tu = z. That is, f and T have a coincidence point.

Since T(E)⊆g(E), Tu = z implies that z ε g(E). Let w εE and gw = z. By using the previous argument, it can be easily verified that Sw = z.

If we assume that g(E) is closed instead of f(E), then we can similarly prove that g and S have a coincidence point.

To prove b, note that {S, g} and {T, f} are weakly compatible and Tu = fu = Sw = gw = z. So, Tz = fz and Sz = gz. Now we show that z is a common fixed point.

Again from 12, we can have:

If in the above inequality, n→∞, since Tz = fz, we obtain:

Hence, φ(d(Tz, z), d(z, Tz)) = 0 and so d(Tz, z) = 0. Therefore, Tz = z and from Tz = fz, we conclude that Tz = fz = z.

Similarly Sz = gz = z. Then, z is a common fixed point of f, g, S and T.

Uniqueness of the common fixed point is a consequence of Eq. 12 and this finishes the proof.

Remark 2a: If in the above theorem, we put the identity map I instead of f and g and E = X, we obtain the theorem 2 of Shatanawi (2011).

Remark 2b: Theorem (7) of Choudhury (2009) is an immediate consequence of the above theorem by taking f = g = 1, T = S and E = X.

Example 2: Let X = R be endowed with the Euclidean metric. Let T, S: X→X be defined by Tx = 1/8 x and Sx = 0, for all x ε X.

We define functions f, g: X→X, : [0, ∞)→[0, ∞) and φ: [0, ∞)2→[0, ∞) by fx = 1/2 x, gx = 2x, Ψ(t) = t/2 and φ(t, s) = t+s/8. Then we have:

Moreover, S and f as well as T and g are weakly compatible, that is, all conditions of theorem 9 are hold. Hence, T, S, f and g have a unique common fixed point (x = 0) by theorem 9.

REFERENCES
1:  Abbas, M., T. Nazir and S. Radenovic, 2011. Common fixed points of four maps in partially ordered metric spaces. Appl. Mathe. Lett., 24: 1520-1526.

2:  Alber, Y.I. and S. Guerre-Delabriere, 1997. Principle of Weakly Contractive Maps in Hilbert Space, New Results in Operator. In: Advances and Applications, Gohberg, I. and Y. Lyubich (Eds.). Vol. 98. Birkhauser, Basel, Switzerland, pp: 7-22.

3:  Beiranvand, A., S. Moradi, M. Omid and H. Pazandeh, 2009. Two fixed point theorem for special mappings. Cornell University Library Gratefully Acknowledge Supporting Institutions. http://arxiv.org/abs/0903.1504.

4:  Chatterjea, S.K., 1972. Fixed point theorems. C.R. Acad. Bulgare Sci., 25: 727-730.

5:  Choudhury, B.S., 2009. Unique fixed point theorem for weak C-contractive mappings. Kathmandu Univ. J. Sci. Eng. Technol., 5: 6-13.

6:  Choudhury, B.S. and A. Kundu, 2012. (ψ-α β) weak contractions in partially ordered metric spaces. Appl. Math. Lett., 25: 6-10.

7:  Harjani, J., B. Lopez and K. Sadarangani, 2011. Fixed point theorems for weakly c-contractive mappings in ordered metric spaces. Comput. Mathe. Appli., 61: 790-796.
CrossRef  |  

8:  Khan, M.S., M. Swaleh and S. Sessa, 1984. Fixed point theorems by altering distancces between the points. Bull. Aust. Mathe. Soc., 30: 1-9.
Direct Link  |  

9:  Nashine, H.K. and B. Samet, 2011. Fixed point results for mappings satisfying (ψ, φ)-weakly contractive condition in partially ordered metric spaces. Nonlinear Anal.: Theory Methods Applic., 74: 2201-2209.
CrossRef  |  Direct Link  |  

10:  Nieto J.J. and R.R. Lopez, 2005. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order, 22: 223-239.
CrossRef  |  

11:  Radenovic S. and Z. Kadelburg, 2010. Generalized weak contractions in partially ordered metric spaces. Comp. Mathe. Applicat., 60: 1776-1783.
CrossRef  |  

12:  Ran, A.C.M. and M.C.B. Reurings, 2004. A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Amer. Mathe. Soc., 132: 1435-1443.

13:  Shakeri, S., M. Jalili, R. Saadati, S.M. Vaezpour and L. Ciric, 2009. Quicksort algorithms: Application of fixed point theorem in probabilistic quasi-metric spaces at domain of words. J. Applied Sci., 9: 397-400.
CrossRef  |  Direct Link  |  

14:  Shatanawi, W., 2011. Fixed point theorems for nonlinear weakly c-contractive mappings in metric spaces. Mathe. Comput. Modell., 54: 2816-2826.
CrossRef  |  

15:  Banach, S., 1922. Sur les operations dans les ensembles abstraits et leurs application aux equations integrales. Fund. Math., 3: 133-181.

16:  Mujahid, A. and D. Dragan, 2010. Common fixed point theorem for four mappings satisfying generalized weak contractive condition. Filomat, 24: 1-10.
Direct Link  |  

17:  Moradi, S., Z. Fathi and E. Analouee, 2011. Common fixed point of single valued generalized φf-weak contractive mappings. Applied Math. Lett., 24: 771-776.
CrossRef  |  Direct Link  |  

18:  Zhang, Q. and Y. Song, 2009. Fixed point theory for generalized φ-weak contractions. Applied Mathematics Lett., 22: 75-78.

19:  Doric, D., 2009. Common fixed point for generalized (ψ, φ)-weak contractions. Applied Math. Lett., 22: 1896-1900.
CrossRef  |  Direct Link  |  

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